Many clinical studies (e.g., cardiovascular outcome trials) investigate the effect of an intervention on multiple event-time outcomes. The most common method of analysis is to conduct a so-called composite" analysis of a composite outcome defined as the time to the first component event. Other approaches have also been proposed, including the win ratio (or win difference) for ordered outcomes and the application of the Wei-Lachin multivariate one-directional test. Herein we assess the influence of the marginal and joint distributions of the component events, and their correlation structures, on the operating characteristics of the composite analysis, and those of other methods for analysis of the multiple events. The operating characteristics are investigated using a bivariate exponential model with a shared frailty, under which these properties can be expressed in closed form. While the composite time to first event analysis provides an unbiased test of the hypothesis of equality of the distribution of the time to first event, we show that it can provide a biased test of the joint null hypothesis of equal marginal hazards when the correlation of event times, or the shared frailty, differs between groups. When the correlations differ between groups, even when the marginal distributions are jointly equal, the probability of significance of the composite analysis can increase substantially. The same applies to the Win Ratio. However, the operating characteristics of the Wei-Lachin or other tests of the joint equality of the marginal hazards are unaffected. In essence, the composite analysis is sensitive to differences in correlations between groups whereas the Wei-Lachin test of joint marginal equality is not. Further, when the correlations are equal, the Wei-Lachin test is more powerful to detect a difference in marginal hazards than is the composite analysis test. Careful consideration and a thorough understanding of the properties of the various methods for analysis of composite outcome measures are in order before adopting one as primary analysis in a clinical study.